A.P. Calculus Fall Midterm

Name _________________________

Directions: Choose the best answer. Calculators allowed. Do not leave any answers blank Good luck. 2x + 10 is x "#5 x + 2x #15

1) lim

2

a) 0

b)

!

"1 8

c)

"1 4

d) ∞

e) none of these

________________________________________________________________________________________ x 3 #1 2) lim 2 is x "1 x #1

a) !

!

3 2

!

b)

1 2

c) 0

d) ∞

e) DNE

________________________________________________________________________________________ ! #3x + 1 is 3) lim 2 x "4 x # 4 ( )

!

a) -11

!

b) -13

c) ∞

d) -∞

e) DNE

________________________________________________________________________________________

$ 4 " x2 & if x # 2 4) Let f ( x ) = % x " 2 &4 if x = 2 '

II. f (2) exists

I. lim f ( x ) exists x "2

!

a) only I

Which of the following statements I,II, and III are true?

b) only II

III. f is continuous at x = 2

c) I and II

d) none of them

e) all of them

! ! ________________________________________________________________________________________ 2 $ & f ( x) = x " 6x for x # 0 5) If % x & f (0) = 2k "1 '

and if f is continuous at x = 0, then k =

!

a) -6

b)

www.MasterMathMentor.com

!

"5 2

c) 0

d)

5 2

e) 6 Stu Schwartz

-1-

!

6)

If f ( x ) = x 2 "1 and g ( x ) =

1 , which of the following statements is FALSE? x

a) f ( x ) is continuous everywhere b) g ( x ) is continuous except at x = 0

! ! !

( ) d) g ( f ( x )) is continuous except at x = 1 c) f g ( x ) is continuous except at x = 0

e) All are true ________________________________________________________________________________________ ! 7) The normal (perpendicular) line to the curve y = 8 " x 2 at (-2, 2) has slope ! a) -2

b)

1 2

c) " !

1 2

d) 1

e) -1

________________________________________________________________________________________ ! 256! + 64 x + 33 x 2 , then f "(64) = 8) If f ( x ) = x a) 4.25

b) 8.75

c) 0.75

d) 10.25

e) 5.78

! ________________________________________________________________________________________ 9)

If the graph of the second derivative of some function f is a line of slope -6, then f could be which type of function? a) constant

b) linear

c) quadratic

d) cubic

e) quartic

________________________________________________________________________________________ 10) If f ( x ) = sin 2 (3 " x ) then f ’(0) = a) -2 cos 3

b) -2 sin 3 cos 3

c) 6 cos 3

d) 2 sin 3 cos 3

e) 6 sin 3 cos 3

!

________________________________________________________________________________________ 11) If f (5) = 3 and f "(5) = #2, find the derivative of x 2 f ( x ) at x = 5. a) 0 ! www.MasterMathMentor.com

b) -18

c) -12 !

-2-

d) -20

e) -80 Stu Schwartz

12)

y ="

a)

! d)

!

1 x2 + 1

, then

dy = dx

x 12

b)

32

e)

( x + 1) 2

"x

( x + 1) 2

!

x

( x + 1) 2

c)

32

x x +1

"x 12

( x + 1) 2

2

!

! ! ________________________________________________________________________________________

13)

If y =

3 dy = , find sin x + cos x dx

a) 3sin x " 3cos x

b)

3(cos x " sin x )

e)

!

!

d)

(sin x + cos x )

2

!

3

(sin x + cos x )

c)

2

"3

(sin x + cos x )

2

3(sin x " cos x ) 1+ 2sin x cos x

!

! ! ________________________________________________________________________________________ dy 14) If x 2 + xy " y = 2, find dx a) !

!

2x + y 1" x

b)

!

2x 1" x

c)

2x " 2 1" x

!

d)

2 " 2x x

e) DNE

!

________________________________________________________________________________________ #" & dy 15) If sin y = cos x , then find at the point % ," ( dx $2 ' " a) -1 b) 0 c) 1 d) e) None of these 2 ! ! ! www.MasterMathMentor.com

-3-

Stu Schwartz

16)

If f ( x ) = x 3 " x, then

3 locates a relative maximum of f 3 3 x= locates a relative minimum of f 3 x = 3 locates a relative maximum of f x = 3 locates a relative minimum of f x = " 3 locates a relative minimum of f

a) x = !

b)

!

c) d) e)

! ! ________________________________________________________________________________________ ! ! f x = x n , where n is a positive integer " 2. The graph of f x will have an inflection point when n is 17)

()

()

a) even

b) odd

c) divisible by 3

d) for all values

e) for no values

! ________________________________________________________________________________________

18)

A particle is subjected to gravity according to the following graph of the particle’s velocity as a function of time t.

At approximately what time t is the particle at its highest point? a) 0 b) 2 c) 3 d) 5 e) 6 ________________________________________________________________________________________ In questions 19-20, a particle moves along a horizontal line according to the formula s = 2t 4 " 4t 3 + 2t 2 "1

19)

The particle is moving right when a) 0 < t <

1! 2

b) t > 0

c) t > 1

1 d) 0 < t < , t > 1 2

e) never

________________________________________________________________________________________ ! ! ! 20) The acceleration, a is increasing when b) t > 0.5

a) t > 1

c) t < 0.211 or t > 0.789

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!

d) 0 < t < 0.5

Stu Schwartz

-4-

!

e) 0 < t < 1

!

!

21)

2" inches per minute. When the radius is 5 inches, 5 how fast is the area of the circle increasing in square inches per minute?

The circumference of a circle is increasing at a rate of

a)

1 5

b)

!

" 5

c) ! 2

d) 2π

e) 25π

!

________________________________________________________________________________________ 22)

A conical tank has a height that is always 3 times its radius. If water is leaving the tank at the rate of 50 cubic feet per minute, how fast is the water level falling in feet per minute when the water is 3 feet high? 1 Volume of a cone is V = "r 2 h . 3 a) 1.000

b) 5.305

c) 15.915

d) 0.589

e) 1.768

!

________________________________________________________________________________________ 23)

A function f ( x ) is continuous for all x and has a local minimum at (1, 8). Which must be true? a) f "(1) = 0 b) f " exists at x = 1 d) f "( x ) < 0 if x < 1, f "( x ) > 0 if x > 1 !

c) the graph is concave up at x = 1 e) f "( x ) > 0 if x < 1, f "( x ) < 0 if x > 1

! ! ! ! ! ! ________________________________________________________________________________________ dy d2y 24) At what point on the graph y = f ( x ) below are both and both less than zero? dx dx 2 a) P b) Q c) R d) S e) T

!

!

! P

S

T

R Q

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-5-

Stu Schwartz

25)

6x 2 1 = , then k = x "# 200 $ 4x $ kx 2 2

If lim a) 3

b) -3

c. 12

d. -12

e. -3

! ________________________________________________________________________________________

26)

For the following piecewise function, determine the answer that best describes it:

$& x " cos( x ), x # 0 f ( x) = % &' x 2 + x "1, x < 0

!

a) both continuous and differentiable

b) neither continuous nor differentiable c) continuous only

d) differentiable only

e) has a cusp point at x = 0

________________________________________________________________________________________ 27)

!

Let f ( x ) be a polynomial function such that f ( 3) = 3, f "( 3) = 0 and f ""( 3) = #3. What is the point (3,3) on the graph y = f ( x ) ? a) relative maximum

!

d) ! inflection point

b) relative minimum

c) intercept

e) none of these

________________________________________________________________________________________ 3 28) Given that f ( x ) = 4 + find all values of c in the interval (1,3) that satisfy the mean value theorem. x a) 2

b)

2

c)

3

d) ± 3

e) MVT doesn’t apply

!

!

!

!

________________________________________________________________________________________ 29)

# (x

2

)

" 4sec x tan x dx =

a) 2x " 4 tan x + C

!

d)

!

x3 " 4 sec x + C 3

www.MasterMathMentor.com

!

b)

x3 " 4 tan x + C 3

c)

x3 " 4 sec 2 x + C 3

e) none of these !

-6-

!

Stu Schwartz

30)

"! % An equation of the line tangent to y = sin x + 2cos x at $ ,1' is #2 & a) 2x " y = # "1 d) 4x + 2 y = 2 " #

b) 2x + y = " + 1 e) none of these

!

! !

c) 2x " 2 y = 2 " #

!

!

________________________________________________________________________________________ 31)

"

x

(

" a) 2$ x #

)

x + 1 dx 3

2

% + x' + C &

b)

x2 + x+C 2

c)

1 2

(

)

2

x +1 + C

3

x 2 2x 2 d) + +C 2 3

!

e) x + 2 x + C !

!

!

! ! ________________________________________________________________________________________ 2

32)

(

a) 0 !

)

For f ( x ) = x 3 x 2 " 4 on [-2 2] the "c" value that satisfies Rolle's Theorem is b) 2

c) ±2

d) There is no value for c because f (0) does not exist e) There is no value for c because f (x) is not differentiable on (-2,2) ! ! !

________________________________________________________________________________________ 33)

!

!

If f (x) is a continuous function with f "( x) = "5x 2 (2x "1)2 (3x + 1)3 , find the set of values of x for which f (x) has an inflection point.

# 1 1& a) $0," , ' 3 2( % !

# 1 1& b) $" !, ' % 3 2(

!

www.MasterMathMentor.com

# 1& c) $" ' % 3(

!

-7-

"1% d) # & $2'

!

e) no inflection points

Stu Schwartz

MMM Midterm TakeHome.pdf

9) If the graph of the second derivative of some function f is a line of slope -6, then f could be which type of. function? a) constant b) linear c) quadratic d) cubic e) ...

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